Here's a a couple of ways I know how to get a usable ID/wallet. I kind of wish that Pickpocketing an ID were a viable method of attaining a wallet, but alas, it is not.
Lockpick method(s):
As long as the time is between 06:00-22:59 , Locker 3-A has a Lockpick in the gym. There are also Grip-Enhancing Gloves that I recommend grabbing and putting on, because they give +5 to your Lockpick skill. (You have to put them on from the "Get changed" menu. There is one in the Locker Room.)
Then, you can either break into Black Hat Detectives and steal Olaf's ID. Or, alternatively, you can attempt to break into Food Processing Center to steal Boy's ID. The latter is significantly harder to break into, and what wallet you get doesn't matter.
Social method:
You can also ask Lilly* to "borrow" Ina's ID. Just approach Lilly, ask about the sign, then ask to borrow her ID. This relies on a Social check, but you can always just save/rely on the autosaves and load to reroll the outcome. (If you aren't above save-scumming, that is. If you are, you can just walk away and come back to ask again.)
*Initially called "Girl holding a sign," on Main Street from 07:00-22:59. If you choose to "Approach people" rather than "Explore the town" at the start of a new game, she will be there. (Despite it being 01:00) You can't walk away and come back, though, because she'll be gone. So if you fail the Social check, you'll have to load the save/autosave, or wait a few in-game hours.